Does putting sugar on minor cut or scrap speed healing?

No: Sugar won't help, but honey will. Honey has natural antibacterial properties. Best is some sort of antibiotic ointment, but if you don't have access to that, then honey works quite well. Hope this helps!

Probably slows it.: I know of no evidence (or even old wive's tale) of sugar speeding up healing. In fact, excess sugar has been shown to SLOW DOWN healing of cuts & wounds (like in diabetics) b/c blood glucose at ~150 mg/dL starts to impair white blood cell mobility. Table sugar is sucrose, but the body breaks it down to glucose & fructose quickly. So, if anything, sugar on minor cuts & scrapes makes healing slower.